Circles Tangents and Chords Objectives To learn 3

Circles, Tangents and Chords Objectives • To learn 3 circle theorems • To use theorems to solve circle questions Keywords Chords, normal © Christine Crisp

Circle Problems Tangents to Circles Some properties of circles may be needed in solving problems. This is the 1 st one Ø The tangent to a circle is perpendicular to the radius at its point of contact A line which is perpendicular to a tangent to any curve is called a normal. x radius tangent For a circle, the radius is a normal.

Circle Problems Tangents to Circles Diagrams are very useful when solving problems involving circles e. g. 1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) Method: The equation of any gradient straight line is. x (5, 7) We need m, the gradient of gradient the tangent. • The Findtangent using (2, 3) x to a circle is tangent perpendicular to the radius at its point of contact • Find m using • Substitute for x, y, and m in to find c.

Circle Problems e. g. 1 Find the equation of the tangent at the point (5, 7) on a circle with centre (2, 3) Solution: gradient x (5, 7) (2, 3) x tangent Substitute the point that is on the tangent, (5, 7): or

Circle Problems e. g. 2 The centre of a circle is at the point C (-1, 2). The radius is 3. Find the length of the tangents from the point P ( 3, 0). Method: Sketch! tangent • Use 1 tangent and join the radius. The required length is AP. C (-1, 2) x • Find CP and use Pythagoras’ theorem for triangle CPA 3 Solution: x tangent P (3, 0) A

Circle Problems Exercises Solutions are on the next 2 slides 1. Find the equation of the tangent at the point A(3, -2) on the circle Ans: 2. Find the equation of the tangent at the point A(7, 6) on the circle Ans:

Circle Problems 1. Find the equation of the tangent at the point A(3, -2) on the circle Solution: Centre is (0, 0). Sketch! Gradient of radius, Gradient of tangent, Equation of tangent is gradient m (0, 0) x gradient or x (3, -2)

Circle Problems 2. Find the equation of the tangent at the point A(7, 6) on the circle Solution: Centre is (4, 2). Gradient of radius, gradient x (7, 6) (4 , 2) x tangent Gradient of tangent, or

Circle Problems Chords of Circles Another useful property of circle is the following: Ø The perpendicular from the centre to a chord bisects the chord x chord

Circle Problems e. g. A circle has equation The point M (4, 3) is the mid-point of a chord. Find the equation of this chord. Method: We need m and c in • x chord • • • Complete the square to find the centre Find the gradient of the radius Find the gradient of the chord Substitute the coordinates of M into to find c.

Circle Problems e. g. A circle has equation The point M (4, 3) is the mid-point of a chord. Find the equation of this chord. Solution: C x chord Centre C is Tip to save time: Could you have got the centre without completing the square?

Circle Problems Exercise 1. A circle has equation (a) Find the coordinates of the centre, C. (b) Find the equation of the chord with midpoint (2, 6). Solution: (a) (b) Centre is ( 1, 5 ) C x chord Equation of chord is on the chord Equation of chord is

Circle Problems Semicircles The 3 rd property of circles that is useful is: Ø The angle in a semicircle is a right angle P B Q x A diameter

Circle Problems e. g. A circle has diameter AB where A is ( -1, 1) and B is (3, 3). Show that the point P (0, 0) lies on the circle. Method: If P lies on the circle the lines AP and BP will be perpendicular. B(3, 3) diameter Solution: x Gradient of AP: A(-1, 1) P(0, 0) Hence Gradient of BP: So, and P is on the circle.

Exercise Circle Problems 1. A, B and C are the points (3, 5), ( -2, 4) and (1, 2) respectively. Show that C lies on the circle with diameter AB. Solution: Gradient of AC A(3, 5) diameter x Gradient of BC B(-2, 4) C(1, 2) Since AC and BC are perpendicular, C lies on the circle diameter AB.

Summary l l l What is the form of an equation of a circle ? What are three circle theorems ? How can we tell if a line intersects a circle, touches it or misses it completely ?